The short answer is nope, dried persimmons don't need to be refrigerated - but, there are some exceptions to the rule. If you live in a scorching hot climate, or you've got a bunch of pesky bugs trying to crash the persimmon party, then yeah, you might want to toss them in the fridge to keep them fresh. But, for the most part, you can just store them in an airtight container and call it a day.
Now, you might be wondering, what's the magic behind dried persimmons? Well, it all comes down to the water content - or, rather, the lack thereof. When persimmons are dried, the water content is reduced to almost zero, making it super hard for bacteria and mold to grow. It's like trying to throw a party in the desert - ain't nobody showing up!
But, don't get too comfy, because there are some caveats to keeping dried persimmons at room temperature. If you've got a humid environment, or you're storing them in a direct sunlight spot, you might want to consider refrigerating them to prevent sogginess or other funky business. And, let's not forget about those pesky critters - if you've got a mouse problem, you'll want to keep those persimmons under lock and key (or, at the very least, in a mouse-proof container).
Dehydrated Persimmon